Projection through quadric mirrors made faster

Abstract

This paper presents a novel framework to project 3D points through curved mirrors to an image device. The problem solved is the search for the reflection point for an arbitrary vision system with a quadric shaped mirror. The main advantage claimed for the framework presented is its computer efficiency while providing better accuracy since the search for the reflection point is made in a parameterized curve that is function of a single unknown. The alternative solvers are the classical Snell Law and the Fermat Principle that, as proved in experiments, present a much slower convergence than the new method since they have a multidimensional search space rather than a unidimensional one. This new method can be used to speed up calibration of noncentral catadioptric systems based on reprojection error. It can also be used for rendering purposes since it enhances the performance of the projection of points through mirrors while enhancing its accuracy, whether the mirror is part of the vision system (catadioptric camera) or if it is only a specular surface in the scene that reflects light in arbitrary directions. Another application of this framework is for illumination purposes, providing a faster way to compute reflected light direction or for the computation of the direction of the light source. Experiments in performance evaluation show the usefulness of the method presented.

Beyond Alhazen's problem: Analytical projection model for non-central catadioptric cameras with quadric mirrors

“…An exact projection model is described by the problem of 'Forward Projection': What is the image projection of a 3D point reflected via a mirror? Previous approaches have acknowledged the difficulty of coming up with an analytical projection model and have resorted to optimization or search based techniques [20,11,19]. We refer to these techniques as iterative forward projection (IFP) [1].…”

Calibrating and centering quasi-central catadioptric cameras

“…In this paper, we combine the AFP model with a perspective camera including lens distortions and derive an approximation that is more than three orders of magnitude faster (i.e., five orders of magnitude with respect to [24]), yet achieves similar accuracy in our experiments. Our results also reveal which parameters are important for modeling axial as well as lateral deviations from the single viewpoint, insights which have been missing in existing comparisons [7].…”

“…Early approaches [21], [9], [22], [23] make use of computationally expensive non-linear optimization and require an initial estimate of the pixel coordinates. Goncalves et al [24] increase efficiency by reducing the optimization to a 1D search problem, but still require ∼ 200 seconds for projecting 10.000 3D points to the image plane.…”

“…Table II shows the time required for projecting 10, 000 random 3D points to the image plane. The running time for the numeric non-central projection has been taken from [24] and is considered approximate. However our results agree with the observations in [10].…”

Fitting line projections in non-central catadioptric cameras with revolution symmetry

“…The forward projection in spherical catadioptric systems is tackled in Agrawal and Ramalingam (2013); Agrawal et al (2011); Gonçalves (2010); Gonçalves and Nogueira (2009). The solution of the forward projection model in this case can be considered as the roots of a polynomial of degree 4.…”